= Solution
<Fourier stability analysis> applies most directly to linear, constant-coefficient evolution schemes on uniform infinite or periodic grids. Translation invariance makes spatial shifts diagonal in <discrete Fourier modes>. If a one-step scheme has $U^{n+1}=\sum_j A_jU^n_{m+j}$, substituting $U_m^n=\widehat U^n(\theta)e^{im\theta}$ gives the <Fourier amplification symbol>
$$
\widehat U^{n+1}(\theta)=G(\theta)\widehat U^n(\theta),
\qquad G(\theta)=\sum_jA_je^{ij\theta}.
$$
The <discrete Parseval identity> identifies the discrete <L2 norm> with the squared norm of Fourier coefficients. Uniform stability on a fixed time interval means a mesh-independent bound on $\|G(\theta)^n\|$ for all frequencies and $n\Delta t\leq T$. In the scalar case $|G|\leq1$ yields contraction. More generally $|G|\leq1+C\Delta t$ gives the bound $e^{CT}$ and is consistent with a continuously growing well-posed equation. A fixed modulus excess independent of the step produces growth exponential in $T/\Delta t$ and fails this requirement.
For systems, eigenvalue moduli alone are insufficient. The <uniform power bound for matrix Fourier symbols> also controls eigenvector conditioning and Jordan growth. For example $G=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ has only unit eigenvalues but $G^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}$, unbounded as $\Delta t\to0$ at fixed $T$. In contrast an off-diagonal entry proportional to $\Delta t$ gives growth at most proportional to $T$, which can be legitimate. A uniformly conditioned diagonalization is a sufficient reduction to scalar modal bounds; <nonnormal matrices> require direct power or energy estimates. Multilevel schemes similarly require the root condition for their frequency-dependent amplification polynomial and a uniform bound on the associated companion evolution, including limiting repeated-root cases.
The simplest parabolic example is the <Forward Euler method> with centered spatial diffusion. For $\mu=\Delta t/h^2$,
$$
G(\theta)=1-4\mu\sin^2(\theta/2),
\qquad \boxed{0\leq\mu\leq\tfrac12}
$$
is the sharp mesh-uniform contraction range. For the <theta method>, $G=[1-4(1-a)\mu\sin^2(\theta/2)]/[1+4a\mu\sin^2(\theta/2)]$, giving unconditional contraction at $a\geq1/2$. Implicit stability permits larger steps, but accuracy still constrains them, and the <Crank-Nicolson method> does not strongly damp high-frequency rough data.
For advection $u_t+cu_x=0$ with $c>0$, the <upwind finite difference scheme> gives
$$
G=1-\nu(1-e^{-i\theta}),\qquad
|G|^2=1-4\nu(1-\nu)\sin^2(\theta/2),
\quad \nu=c\Delta t/h.
$$
Thus $0\leq\nu\leq1$ is its contraction range. The <Lax-Friedrichs scheme> has $G=\cos\theta-i\nu\sin\theta$ and is contractive exactly when $|\nu|\leq1$. These express a <Courant–Friedrichs–Lewy condition>: the numerical domain of dependence must encompass the physical one for convergence. That necessary condition is not sufficient by itself. Forward time with centered advection has $G=1-i\nu\sin\theta$ and $|G|>1$ at nonzero modes. At a fixed nonzero Courant number it is unstable even if $|\nu|\leq1$. Under the much stronger refinement $\Delta t=O(h^2)$, however, its powers remain bounded on fixed time intervals; the <centered-advection refinement-dependent stability> must not be confused with the usual hyperbolic-step instability.
Symbols also quantify dispersion and dissipation. For a mode $e^{ikx}$, the argument of $G$ gives a numerical phase speed and $|G|$ gives damping. Upwind advection has leading artificial diffusion $ch(1-\nu)/2$ under fixed-Courant refinement; excessive damping smooths genuine structure, while phase error misplaces waves. Stability controls boundedness, not either error by itself. Smooth low-frequency <Taylor expansions> establish consistency; high-frequency modal bounds establish stability, so analyzing only fixed physical wavenumbers can miss grid-scale instabilities.
For a well-posed linear initial-value problem, the <Lax equivalence theorem> makes consistency plus a uniform stability estimate equivalent to convergence in the chosen norm. Boundaries and variable coefficients need extra care: Fourier diagonalization no longer handles the full update, and an interior or frozen-coefficient symbol does not prove boundary or global stability. Boundary energy estimates, a suitable matrix analysis or a separate boundary-stability argument must supplement it. Nonlinear schemes also require controls beyond this linear symbol test. These limitations delimit what the method proves rather than reducing its usefulness: in its proper setting, it turns an arbitrarily large grid evolution into exact and interpretable modal algebra.
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